Inverse problems are ubiquitous and statistical methods for their treatment have been developed for a long time [1]}, [2]}, [3]}.
Frequently, inverse problems are considered in a discretized form, resulting in (large-scale) regression tasks, or they are posed as discrete inverse problems from the start, for example in functional MR imaging [4]}, [5]}. Inverse problems are often ill-posed, or ill-conditioned in the discrete case, and reliable estimation requires some form of regularization such as Tikhonov regularization [6]}. Bayesian inference [7]}, [8]} provides an alternative approach to ill-posed inverse problems, in which the employed prior renders the estimation task well-posed. For example, a Gaussian prior can lead to a maximum a posteriori (MAP) estimate equivalent to the result of a Tikhonov regularization. Gaussian Markov random field (GMRF) priors [9]} are another popular class of priors used. For example, to model a priori spatial smoothness as it is often relevant in spatial modeling or image processing.
